
where the last line follows by the mean value theorem. We now deal with the last two terms in the above expression.
By corollary 2.2.2 in [1]}, \( ({k}/{N})Y_{N-k, N}\xrightarrow{} 1\) .
By corollary 1.1.10 in [1]} under assumptions REF  and REF  it holds that
\(\lim _{t\rightarrow \infty } \dfrac{U^{\prime }_F(tx)}{U^{\prime }_F(t)} = x^{\gamma -1}\) 
